We propose the systematic use of Hermitian symmetric spaces in representation learning. Symmetric spaces are Riemannian manifolds with rich symmetry groups. This makes them amenable to analysis and explicit computations.
Hermitian symmetric spaces are modeled over the complex numbers \( and carry an invariant complex structure. Furthermore, they admit matrix models of the Poincaré disc and the upper half-plane model of hyperbolic space in \) N\(. This has several mathematical advantages, as it leads to a simple expression for the gradient of a function, and provides new analytical tools and a well-established theory of harmonic analysis.\)Matrix models: We focus on the Siegel spaces \(\mathcal {H}_n\)  [1]} as examples of Hermitian symmetric spaces. Siegel spaces generalize the hyperbolic plane which is \(\mathcal {H}_1\) . They admit a bounded domain model and an upper half-space model in the space \({\rm Sym}(n,\)  of \(n \times n\)  symmetric matrices with complex entries. When \(n=1\) , these are the Poincaré model and the upper half-space model of the hyperbolic plane. The dimension of the space \(\mathcal {H}_n\)  is \(n(n+1)\) , and it contains \(n\) -dimensional Euclidean subspaces, products of \(n\) -copies of hyperbolic planes, as well as products of Euclidean and hyperbolic spaces.
